Properties and Application of the Location‐Test V (n, k)

The present paper is concerned with the properties of a test statistic V ( n, k ) to test location differences in the one‐sample case with known hypothetical distribution G ( x ). The test is similar to the W ILCOXON two‐sample statistic after replacement of the second sample by quantiles of the hypothetical distribution. A comparison with the exact distribution of V ( n, k ) shows that an approximation by means of the normal distribution provides good results even for small sample sizes. The V ‐test is unbiased against one‐tailed alternatives and it is consistent with a restriction which is hardly relevant in practical applications. With regard to the application we are interested especially in the power and robustness against extreme observations for small sample size n . It is shown that in a normal distribution with known standard deviation V ( n, k ) is more powerful than S TUDENT's t for small n and more robust in the sense considered here. The test statistic is based on grouping of the observations into classes of equal expected frequency. A generalization to arbitrary classes provides an essential extension of applicability such as to discrete distributions and to situations where only relative frequencies of G ( x ) in fixed classes are known.